Problem: $p(t) = (4t - 3t^2, t + 5, t^8 - 4t^2)$ What is the velocity of $p(t)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(-6t, 0, 56t^6 - 8)$ (Choice B) B $(4 - 6t, 1, 8t^7 - 8t)$ (Choice C) C $\sqrt{64t^{14} - 128t^8 + 100t^2 - 48t + 16}$ (Choice D) D $\sqrt{32t^{14} + 128t^8 + 36t^2 - 48t - 16}$
Solution: The velocity of a parametric curve is the derivative of its position. If $f(t) = (a(t), b(t), c(t))$, then the velocity is: $f'(t) = (a'(t), b'(t), c'(t))$ Our position function here is $p(t)$. $p'(t) = (4 - 6t, 1, 8t^7 - 8t)$ Therefore, the velocity of $p(t)$ is $(4 - 6t, 1, 8t^7 - 8t)$.